Three Levers

The scoring function has three parameters:

score_i(x) = log(b_i) - ||x - c_i||² / σ_i²

σ Is the Advertiser’s Lever

A climbing physical therapist sets σ tight: only queries about climbing injuries and finger pulley rehab. A general sports PT sets σ wide. Anything sports-related is worth a bid.

The scoring function already rewards accurate positioning: an advertiser closer to the query wins, all else equal. σ controls how much of the space around that position they’re willing to serve. At the center the score is log(b) regardless of σ; what σ sets is how fast the score decays from there, and how much of what you win you can actually convert. Tight σ spends nothing outside the niche. Wide σ keeps competing far from center and pays for everything it takes.

τ Is the Publisher’s Lever

Before bids are considered, the publisher applies a relevance threshold τ. Only ads whose distance to the content falls below τ enter the auction. This is a UX decision. The publisher is saying “ads on my site must be at least this relevant.”

τ is a knob, tuned by directional feedback. Tighten it if users are bouncing; loosen it if fill rates are too low.

Hartline, Hoy & Taggart (2023) prove that competitive efficiency is closed under reserve pricing. τ has zero auction cost.

τ is zoning. The publisher decides what’s allowed in the neighborhood.

Is log(b) the Platform’s Lever?

There’s a third term in the scoring function. What does it do?

Consider a publisher who sets τ = ∞. No relevance gate at all. Without log compression, the auction becomes highest-bidder-wins. A $100 bid stomps a $7 bid regardless of fit.

Log compression caps that. With log scoring:

• log(100) − log(7) ≈ 2.66
• log(7) − log(2) ≈ 1.25

The $100 bidder gets a 2.66-unit head start over the $7 bidder. The distance term ||x − c||² / σ² easily swings by 3–5 units between a well-matched ad and a mediocre one. Distance still decides most auctions, even with no τ.

Each lever reshapes the power diagram differently: σ stretches individual cells, τ clips them at a radius, log(b) shifts boundaries with diminishing returns. Two levers and a candidate.

Information Asymmetry

Advertisers publish their embeddings. Users and publishers don’t.

Advertiser embeddings are public. “I want people interested in marathon training” is a storefront sign. Advertisers can see each other’s positions, like seeing what shelf space a competitor occupies.

Publisher and user embeddings are private. The auction runs inside the publisher’s domain or a trusted execution environment. Advertisers learn only whether their ad passed and at what price. The publisher can verify the auction because all advertiser embeddings are public inputs. The user’s query never leaves the gate.

Advertisers learn where demand is indirectly, through performance feedback, the way a store learns from foot traffic.

Platform Competition Does the Rest

Who forces the platform to serve the user’s interests?

For LLM platforms, the answer is competition. Switching from ChatGPT to Claude to Perplexity costs nothing. No data lock-in, no switching penalty. If a platform over-serves ads or lets irrelevant ones through, users leave.

Perplexity already demonstrated this: total ad revenue came in at $20,000 against $34 million in subscriptions. They killed the ad program. The market revealed that for their users, ad-free was worth paying for, and ads weren’t worth the churn risk.

Getting τ wrong is fatal: the platform that shows irrelevant ads loses users to one that gets it right. Competition audits the levers continuously.

Which raises a question about log(b). Whatever compression the platform applies, advertisers adjust σ. The base enters the allocation only through one combination: multiply the score through by ln(b) and every auction ranks ln(bid) - ln(b)·distance²/σ². Replace σ with σ·√ln(b) and every score, every boundary, every payment lands exactly where it was. The log-base sweep even implements its sweep as that substitution, and it holds σ fixed, which is why it measures movement at all: the dial shifts allocation only until advertisers finish rescaling. So log(b) protects no one and earns no rent at equilibrium. It reads the same market in different units.

The lever’s value is in the turning. An exchange that controls the compression function shifts allocation only transiently, during the lag while σ adapts. The durable rents are the real-estate kind: the government takes its cut through property taxes, agents take theirs on the switch. Here both collectors are the platform. Auction payments tax the sitting; relocation fees price the move. The scoring function has two levers and a market.

Editor’s note: I originally read log(b) as the platform’s lever, the third of the three in the title. The σ-compensation identity above is what changed my mind. The formalization tracks the equilibrium version, that advertisers do rescale, as an open claim pending σ best-response theory; the identity itself is arithmetic.

Part of the Vector Space series.